Finite Group Databasebeta

← #1860×C211 #41009480

Subgroups — #1860×C211 #41009480

Order 27008.

286 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#520×C211C211yesmaximal, characteristic
2#428×C211C211yesmaximal, characteristic, complemented
3C2×#104×C211C211yesmaximal, characteristic
4#417×C211C211yesmaximal, characteristic
5#382×C211C211yesmaximal, characteristic, complemented
6#343×C211C211yesmaximal, characteristic, complemented
7#339×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211C2211yescharacteristic
9C2×M16×C211C2211yescharacteristic
10C2×M16×C211C2211yescharacteristic
11C4×Q8×C211—21no—
12C4×D4×C211—21no—
13C4×D4×C211—22no—
14#117×C211—21no—
15#117×C211C2211yescharacteristic
16C2×C4⋊C4×C211—21no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C42×C211C2211yescharacteristic
20#107×C211—21no—
21#106×C211—21no—
22#104×C211C414yescomplemented
23#103×C211—21no—
24#102×C211—21no—
25#95×C211—22no—
26#41×C211—21no—
27#41×C211—21no—
28#41×C211C2×C412yes—
29#41×C211Q812yes—
30C2×Q8×C211D411yescharacteristic
31C2×D4×C211—21no—
32C2×D4×C211D411yescharacteristic
33C22×C4×C211—21no—
34C22×C4×C211—21no—
35C22×C4×C211—22no—
36C22×C4×C211—21no—
37C22×C4×C211D412yes—
38C22×C4×C211C2311yescharacteristic
39M16×C211—21no—
40M16×C211—21no—
41M16×C211—21no—
42M16×C211C2×C412yescomplemented
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—22no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211D411yescharacteristic
51C4⋊C4×C211D411yescharacteristic
52C22⋊C4×C211—22no—
53C22⋊C4×C211—21no—
54C42×C211—22no—
55C42×C211—21no—
56C42×C211—21no—
57C42×C211C2×C412yescomplemented
58C23×C211—21no—
59C23×C211C2×D411yescharacteristic
60Q8×C211—21no—
61Q8×C211C4⋊C412yes—
62D4×C211—22no—
63D4×C211—22no—
64D4×C211—21no—
65D4×C211C4⋊C412yes—
66C2×C4×C211—21no—
67C2×C4×C211—22no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—21no—
71C2×C4×C211—22no—
72C2×C4×C211—21no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—42no—
76C2×C4×C211—42no—
77C2×C4×C211—41no—
78C2×C4×C211—82no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×Q811yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—41no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—22no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211C4×D411yescharacteristic
93C22×C211#12011yescharacteristic
94C22×C211C22⋊Q811yescharacteristic
95C4×C211—22no—
96C4×C211—42no—
97C4×C211—41no—
98C4×C211—42no—
99C4×C211—41no—
100C4×C211—81no—
101C4×C211C2×C4⋊C411yescharacteristic
102C4×C211C4×D411yescharacteristic
103C4×C211C22⋊Q811yescharacteristic
104C4×C211#12311yescharacteristic
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—42no—
108C2×C211#38511yescharacteristic
109C2×C211#45411yescharacteristic
110C2×C211#45511yescharacteristic
111C211#186011yescharacteristic, direct product factor, Sylow
112#1860C21111yesmaximal, characteristic, direct product factor, Sylow
113#520C2×C21111yescharacteristic
114#428C2×C21111yescharacteristic, complemented
115C2×#104C2×C21111yescharacteristic
116#417C2×C21111yescharacteristic
117#382C2×C21111yescharacteristic, complemented
118#343C2×C21111yescharacteristic, complemented
119#339C2×C21111yescharacteristic, complemented
120C2×#41C22×C21111yescharacteristic
121C2×M16C22×C21111yescharacteristic
122C2×M16C22×C21111yescharacteristic
123C4×Q8—21no—
124C4×D4—21no—
125C4×D4—22no—
126#117—21no—
127#117C22×C21111yescharacteristic
128C2×C4⋊C4—21no—
129C2×C4⋊C4C22×C21111yescharacteristic
130C2×C4⋊C4C22×C21111yescharacteristic
131C2×C42C22×C21111yescharacteristic
132#107—21no—
133#106—21no—
134#104C4×C21114yescomplemented
135#103—21no—
136#102—21no—
137#95—22no—
138#41—21no—
139#41—21no—
140#41C2×C4×C21112yes—
141#41Q8×C21112yes—
142C2×Q8D4×C21111yescharacteristic
143C2×D4—21no—
144C2×D4D4×C21111yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4D4×C21112yes—
150C22×C4C23×C21111yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16C2×C4×C21112yescomplemented
155C2×C8—21no—
156C2×C8—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4D4×C21111yescharacteristic
163C4⋊C4D4×C21111yescharacteristic
164C22⋊C4—22no—
165C22⋊C4—21no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42C2×C4×C21112yescomplemented
170C23—21no—
171C23C2×D4×C21111yescharacteristic
172Q8—21no—
173Q8C4⋊C4×C21112yes—
174D4—22no—
175D4—22no—
176D4—21no—
177D4C4⋊C4×C21112yes—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4C22×C4×C21111yescharacteristic
192C2×C4C2×D4×C21111yescharacteristic
193C2×C4C2×D4×C21111yescharacteristic
194C2×C4C2×Q8×C21111yescharacteristic
195C2×C4#41×C21111yescharacteristic
196C2×C4#41×C21111yescharacteristic
197C8—22no—
198C8—41no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22C4×D4×C21111yescharacteristic
205C22#120×C21111yescharacteristic
206C22C22⋊Q8×C21111yescharacteristic
207C4—22no—
208C4—42no—
209C4—41no—
210C4—42no—
211C4—41no—
212C4—81no—
213C4C2×C4⋊C4×C21111yescharacteristic
214C4C4×D4×C21111yescharacteristic
215C4C22⋊Q8×C21111yescharacteristic
216C4#123×C21111yescharacteristic
217C2—21no—
218C2—21no—
219C2—42no—
220C2#385×C21111yescharacteristic
221C2#454×C21111yescharacteristic
222C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#41×C211C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #428×C211C2semidirect
5. #382×C211C2semidirect
6. #343×C211C2semidirect
7. #339×C211C2semidirect
22. #104×C211C4semidirect
42. M16×C211C2×C4semidirect
57. C42×C211C2×C4semidirect
111. C211#1860direct (N × H)
112. #1860C211direct (N × H)
114. #428C2×C211semidirect
117. #382C2×C211semidirect
118. #343C2×C211semidirect
119. #339C2×C211semidirect
134. #104C4×C211semidirect
154. M16C2×C4×C211semidirect
169. C42C2×C4×C211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#520×C2112
#428×C2112
C2×#104×C2112
#417×C2112
#382×C2112
#343×C2112
#339×C2112
#1860211